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Interference

# Interference - Mathematical Method Electric fields at point...

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Liu UCD Phy9B 07 1 Ch 35. Interference

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Liu UCD Phy9B 07 2 35-1. Interference & Coherence Sources c= λ f
Liu UCD Phy9B 07 3 Interference Path difference λ /2, 3 λ /2, 5 λ /2… Destructive interference 0, λ, 2λ, 3λ... Constructive interference Coherent Sources: two monochromatic sources of the same frequency & with any definite, constant phase relation (not necessarily in phase). r=2 λ Constructive r=2.5 λ Destructive

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Liu UCD Phy9B 07 4 35-2. Two Source Interference of Light Thomas Young’s experiment (1800) Assumptions: Monochromatic Coherent Path difference r 2 -r 1 =d sin θ Constructive interference (Bright fringes): Destructive interference (Dark fringes): ,... 2 , 1 , 0 ± ± = m λ θ m d = sin λ θ ) 2 1 ( sin + = m d
Liu UCD Phy9B 07 5 Interference Fringes For small angles only Constructive interference in Young’s Exp: y max =R tan θ m Rsin θ m y max = R m λ /d =0, ±R λ /d, ±2R λ /d, ±3R λ /d… Center is a maximum y min = R( m + 1/2) λ /d = ±R λ /2d, ±3R λ /2d, ±5R λ /2d… Spacing between adjacent maxima /minima: R λ /d (R>>d, R>>y m )

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Liu UCD Phy9B 07 6 35-3. Intensity in Interference Patterns:

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Unformatted text preview: Mathematical Method Electric fields at point P: E 1 =E cos ( ω t+ φ ) E 2 =E cos t Superposition: E 1 + E 2 =E cos ( t+ )+ E cos t = 2E cos ( φ/2) cos ( t+ φ/2) = Ε P cos ( t+ Amplitude: P = 2E | cos ( φ/2)| Intensity I ∝ E P 2 = 4E 2 cos 2 ( Or: I=I o cos 2 ( Liu UCD Phy9B 07 7 Intensity in Interference Patterns: Phasor Diagram Amplitude: Ε P = 2E | cos ( φ/2)| Intensity I ∝ E P 2 = 4E 2 cos 2 ( φ/2) Or: I=I o cos 2 ( φ/2) Liu UCD Phy9B 07 8 Intensity at y λ π φ 1 2 2 r r − = Phase difference: Intensity: θ sin 2 sin ) ( ) ( 2 1 2 1 2 d kd r r k r r = = − = − = ) sin ( cos 2 cos 2 2 d I I I o o = = Maximum Intensity: or: Small slits ( y<<R, then sin θ = y/R ) m d = sin m d = sin ) ( cos 2 R dy I I o = All peaks have same intensity....
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